INFO-I 421 Final - Building Decision Tree Models for the Breast Cancer Data
Nicholas Bibeau
April 29, 2019
1.1 Importing Data
breast_cancer <- as_tibble(read.table("https://archive.ics.uci.edu/ml/machine-learning-databases/breast-cancer/breast-cancer.data", header = FALSE, na = "?", sep = ",", col.names=c("class", "age", "menopause", "tumor_size", "inv_nodes", "node_caps", "deg_malig", "breast", "breast_quad", "irradiat")))1.2 Finding the age category which has the highest risk of recurrence
Viewing distinct factors for age
unique(breast_cancer$age)## [1] 30-39 40-49 60-69 50-59 70-79 20-29
## Levels: 20-29 30-39 40-49 50-59 60-69 70-79Building contingency table for age and recurrence
breast_cancer_table <- table(breast_cancer$age, breast_cancer$class, dnn = c("Age", "Recurrence"))
breast_cancer_table## Recurrence
## Age no-recurrence-events recurrence-events
## 20-29 1 0
## 30-39 21 15
## 40-49 63 27
## 50-59 71 25
## 60-69 40 17
## 70-79 5 1Calculating probability of recurrence by age group
probAge1 <- count(breast_cancer %>% filter(class == "recurrence-events") %>% filter(age == "20-29")) /
count(breast_cancer %>% filter(age == "20-29"))
probAge2 <- count(breast_cancer %>% filter(class == "recurrence-events") %>% filter(age == "30-39")) /
count(breast_cancer %>% filter(age == "30-39"))
probAge3 <- count(breast_cancer %>% filter(class == "recurrence-events") %>% filter(age == "40-49")) /
count(breast_cancer %>% filter(age == "40-49"))
probAge4 <- count(breast_cancer %>% filter(class == "recurrence-events") %>% filter(age == "50-59")) /
count(breast_cancer %>% filter(age == "50-59"))
probAge5 <- count(breast_cancer %>% filter(class == "recurrence-events") %>% filter(age == "60-69")) /
count(breast_cancer %>% filter(age == "60-69"))
probAge6 <- count(breast_cancer %>% filter(class == "recurrence-events") %>% filter(age == "70-79")) /
count(breast_cancer %>% filter(age == "70-79"))
# Creating vector of recurrence probability by age group
recur_probs <- c(probAge1, probAge2, probAge3, probAge4, probAge5, probAge6)
# Creating vector of distinct age groups
ages <- c ("20-29", "30-39", "40-49", "50-59", "60-69", "70-79")
# Creating named vector of recurrence probability by age group
names(recur_probs) <- agesDetermining age category with the highest risk of recurrence
# The age group with the highest risk of recurrence is:
names(recur_probs)[which.max(recur_probs)]## [1] "30-39"1.3 Finding breast with the higher risk of recurrence
Viewing distinct factors for breast
unique(breast_cancer$breast)## [1] left right
## Levels: left rightBuilding contingency table for age and recurrence
breast_cancer_table <- table(breast_cancer$breast, breast_cancer$class, dnn = c("Breast", "Recurrence"))
breast_cancer_table## Recurrence
## Breast no-recurrence-events recurrence-events
## left 103 49
## right 98 36Calculating probability of recurrence by breast
probLeft <- count(breast_cancer %>% filter(class == "recurrence-events") %>% filter(breast == "left")) /
count(breast_cancer %>% filter(breast == "left"))
probRight <- count(breast_cancer %>% filter(class == "recurrence-events") %>% filter(breast == "right")) /
count(breast_cancer %>% filter(breast == "right"))
# Creating vector of recurrence probability by breast
recur_probs_breasts <- c(probLeft, probRight)
# Creating vector of distinct breasts
breasts <- c ("left", "right")
# Creating named vector of recurrence probability by age group
names(recur_probs_breasts) <- breastsDetermining breast with the higher risk of recurrence
# The age group with the highest risk of recurrence is:
names(recur_probs_breasts)[which.max(recur_probs_breasts)]## [1] "left"1.4 Drop missing values
breast_cancer <- breast_cancer %>% drop_na()1.5 Rank attributes and plot correlation matrix
Building correlation matrix
# All vectors in breast_cancer dataset are numeric or factors, so no mutations are necessary.
breast_cancer.cor <- hetcor(breast_cancer$class, breast_cancer$age, breast_cancer$menopause, breast_cancer$tumor_size, breast_cancer$inv_nodes, breast_cancer$node_caps, breast_cancer$deg_malig, breast_cancer$breast, breast_cancer$breast_quad, breast_cancer$irradiat)## Warning in log(P): NaNs produced
## Warning in log(P): NaNs produced
## Warning in log(P): NaNs produced
## Warning in log(P): NaNs produced
## Warning in log(P): NaNs produced
## Warning in log(P): NaNs produced
## Warning in log(P): NaNs producedStoring correlation matrix with updated column names
# Creating correlation matrix
breast_cancer.corrmat <- as_tibble(breast_cancer.cor$correlations)
# Assigning original column names to correlation matrix
colnames(breast_cancer.corrmat) <- colnames(breast_cancer)
# Viewing correlation matrix
breast_cancer.corrmat## # A tibble: 10 x 10
## class age menopause tumor_size inv_nodes node_caps deg_malig
## <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 1 -0.123 0.0995 0.222 0.505 0.481 0.425
## 2 -0.123 1 -0.884 -0.0214 -0.143 -0.0703 -0.0711
## 3 0.0995 -0.884 1 0.00476 0.0445 0.0455 -0.0588
## 4 0.222 -0.0214 0.00476 1 0.268 0.274 0.177
## 5 0.505 -0.143 0.0445 0.268 1 0.852 0.444
## 6 0.481 -0.0703 0.0455 0.274 0.852 1 0.495
## 7 0.425 -0.0711 -0.0588 0.177 0.444 0.495 1
## 8 -0.0686 -0.0537 0.0872 0.0316 0.0462 -0.0221 -0.0577
## 9 0.0794 -0.0305 0.0955 0.185 0.0728 0.0936 0.0523
## 10 0.381 -0.109 0.126 0.244 0.530 0.548 0.332
## # ... with 3 more variables: breast <dbl>, breast_quad <dbl>,
## # irradiat <dbl>Adding attribute names in a new column and reordering columns
breast_cancer.corrmat <- breast_cancer.corrmat %>%
mutate(attribute=colnames(breast_cancer.corrmat)) %>%
select(attribute, class, age, menopause, tumor_size, inv_nodes, node_caps, deg_malig, breast, breast_quad, irradiat)Printing the ranking
rank <- breast_cancer.corrmat %>%
select(attribute, class) %>%
mutate(squared.correlation=class^2) %>%
arrange(desc(squared.correlation))
rank## # A tibble: 10 x 3
## attribute class squared.correlation
## <chr> <dbl> <dbl>
## 1 class 1 1
## 2 inv_nodes 0.505 0.255
## 3 node_caps 0.481 0.232
## 4 deg_malig 0.425 0.180
## 5 irradiat 0.381 0.146
## 6 tumor_size 0.222 0.0495
## 7 age -0.123 0.0150
## 8 menopause 0.0995 0.00991
## 9 breast_quad 0.0794 0.00631
## 10 breast -0.0686 0.00471Visualizing the correlation matrix in a correlogram
corrplot(breast_cancer.cor$correlations, method="shade", type="upper", tl.col="black", tl.srt=45)
1.6 Decision tree models
1.6.1 Calculation information gain for each regular attribute
Computing entropy of dataset for gain calculations
#Finding counts for entropy calculation
counts <- (breast_cancer %>%
select(class) %>%
group_by(class) %>%
summarise(count=n())
)$count
# Calculating entropy and gini
entropy_set <-entropy.gini(counts)
# The entropy for the entire dataset is:
entropy_set$entropy## [1] 1Calculating gain based on age
Looking at distribution in terms of three subsets by age
info.before <- entropy_set$entropy
subsets.age <- table(breast_cancer$age, breast_cancer$class)
subsets.age##
## no-recurrence-events recurrence-events
## 20-29 1 0
## 30-39 21 15
## 40-49 62 27
## 50-59 69 22
## 60-69 38 17
## 70-79 5 0Computing entropy of age subsets
entropy20_29 <-entropy.gini(subsets.age["20-29",], type=2)$entropy
entropy30_39 <-entropy.gini(subsets.age["30-39",], type=2)$entropy
entropy40_49 <-entropy.gini(subsets.age["40-49",], type=2)$entropy
entropy50_59 <-entropy.gini(subsets.age["50-59",], type=2)$entropy
entropy60_69 <-entropy.gini(subsets.age["60-69",], type=2)$entropy
entropy70_79 <-entropy.gini(subsets.age["70-79",], type=2)$entropy
Computing weighted entropy for age
total <- nrow(breast_cancer)
w20_29 <- sum(subsets.age["20-29",])/total
w30_39 <- sum(subsets.age["30-39",])/total
w40_49 <- sum(subsets.age["40-49",])/total
w50_59 <- sum(subsets.age["50-59",])/total
w60_69 <- sum(subsets.age["60-69",])/total
w70_79 <- sum(subsets.age["70-79",])/total
info.age <- w20_29*entropy20_29 + w30_39*entropy30_39 + w40_49*entropy40_49 + w50_59*entropy50_59 + w60_69*entropy60_69 + w70_79*entropy70_79Computing information gain for age
gain.age <- info.before - info.age
gain.age## [1] 0.1489036Calculating gain based on menopause
Looking at distribution in terms of three subsets by menopause
subsets.menopause <- table(breast_cancer$menopause, breast_cancer$class)
subsets.menopause##
## no-recurrence-events recurrence-events
## ge40 90 33
## lt40 5 0
## premeno 101 48Computing entropy of menopause subsets
entropy_premeno <-entropy.gini(subsets.menopause["premeno",], type=2)$entropy
entropy_ge40 <-entropy.gini(subsets.menopause["ge40",], type=2)$entropy
entropy_lt40 <-entropy.gini(subsets.menopause["lt40",], type=2)$entropy
Computing weighted entropy for menopause
w_premeno <- sum(subsets.menopause["premeno",])/total
w_ge40 <- sum(subsets.menopause["ge40",])/total
w_lt40 <- sum(subsets.menopause["lt40",])/total
info.menopause <- w_premeno*entropy_premeno + w_ge40*entropy_ge40 + w_lt40*entropy_lt40Computing information gain for menopause
gain.menopause <- info.before - info.menopause
gain.menopause## [1] 0.1397255Calculating gain based on tumor size
Looking at distribution in terms of three subsets by tumor size
subsets.tumor_size <- table(breast_cancer$tumor_size, breast_cancer$class)
subsets.tumor_size##
## no-recurrence-events recurrence-events
## 0-4 7 1
## 10-14 27 1
## 15-19 23 6
## 20-24 34 14
## 25-29 33 18
## 30-34 33 24
## 35-39 12 7
## 40-44 16 6
## 45-49 2 1
## 5-9 4 0
## 50-54 5 3Computing entropy of tumor size subsets
entropy_ts01 <-entropy.gini(subsets.tumor_size["0-4",], type=2)$entropy
entropy_ts02 <-entropy.gini(subsets.tumor_size["5-9",], type=2)$entropy
entropy_ts03 <-entropy.gini(subsets.tumor_size["10-14",], type=2)$entropy
entropy_ts04 <-entropy.gini(subsets.tumor_size["15-19",], type=2)$entropy
entropy_ts05 <-entropy.gini(subsets.tumor_size["20-24",], type=2)$entropy
entropy_ts06 <-entropy.gini(subsets.tumor_size["25-29",], type=2)$entropy
entropy_ts07 <-entropy.gini(subsets.tumor_size["30-34",], type=2)$entropy
entropy_ts08 <-entropy.gini(subsets.tumor_size["35-39",], type=2)$entropy
entropy_ts09 <-entropy.gini(subsets.tumor_size["40-44",], type=2)$entropy
entropy_ts10 <-entropy.gini(subsets.tumor_size["45-49",], type=2)$entropy
entropy_ts11 <-entropy.gini(subsets.tumor_size["50-54",], type=2)$entropy
Computing weighted entropy for tumor_size
w_ts01 <- sum(subsets.tumor_size["0-4",])/total
w_ts02 <- sum(subsets.tumor_size["5-9",])/total
w_ts03 <- sum(subsets.tumor_size["10-14",])/total
w_ts04 <- sum(subsets.tumor_size["15-19",])/total
w_ts05 <- sum(subsets.tumor_size["20-24",])/total
w_ts06 <- sum(subsets.tumor_size["25-29",])/total
w_ts07 <- sum(subsets.tumor_size["30-34",])/total
w_ts08 <- sum(subsets.tumor_size["35-39",])/total
w_ts09 <- sum(subsets.tumor_size["40-44",])/total
w_ts10 <- sum(subsets.tumor_size["45-49",])/total
w_ts11 <- sum(subsets.tumor_size["50-54",])/total
info.tumor_size <- w_ts01*entropy_ts01 + w_ts02*entropy_ts02 + w_ts03*entropy_ts03 + w_ts04*entropy_ts04 + w_ts05*entropy_ts05 + w_ts06*entropy_ts06 + w_ts07*entropy_ts07 + w_ts08*entropy_ts08 + w_ts09*entropy_ts09 + w_ts10*entropy_ts10 + w_ts11*entropy_ts11Computing information gain for tumor_size
gain.tumor_size <- info.before - info.tumor_size
gain.tumor_size## [1] 0.1896315Calculating gain based on inv_nodes
Looking at distribution in terms of three subsets by inv_nodes
subsets.inv_nodes <- table(breast_cancer$inv_nodes, breast_cancer$class)
subsets.inv_nodes##
## no-recurrence-events recurrence-events
## 0-2 166 43
## 12-14 1 2
## 15-17 3 3
## 24-26 0 1
## 3-5 17 17
## 6-8 7 10
## 9-11 2 5Computing entropy of inv_nodes subsein
entropy_in01 <-entropy.gini(subsets.inv_nodes["0-2",], type=2)$entropy
entropy_in02 <-entropy.gini(subsets.inv_nodes["3-5",], type=2)$entropy
entropy_in03 <-entropy.gini(subsets.inv_nodes["6-8",], type=2)$entropy
entropy_in04 <-entropy.gini(subsets.inv_nodes["9-11",], type=2)$entropy
entropy_in05 <-entropy.gini(subsets.inv_nodes["12-14",], type=2)$entropy
entropy_in06 <-entropy.gini(subsets.inv_nodes["15-17",], type=2)$entropy
entropy_in07 <-entropy.gini(subsets.inv_nodes["24-26",], type=2)$entropy
Computing weighted entropy for inv_nodes
w_in01 <- sum(subsets.inv_nodes["0-2",])/total
w_in02 <- sum(subsets.inv_nodes["3-5",])/total
w_in03 <- sum(subsets.inv_nodes["6-8",])/total
w_in04 <- sum(subsets.inv_nodes["9-11",])/total
w_in05 <- sum(subsets.inv_nodes["12-14",])/total
w_in06 <- sum(subsets.inv_nodes["15-17",])/total
w_in07 <- sum(subsets.inv_nodes["24-26",])/total
info.inv_nodes <- w_in01*entropy_in01 + w_in02*entropy_in02 + w_in03*entropy_in03 + w_in04*entropy_in04 + w_in05*entropy_in05 + w_in06*entropy_in06 + w_in07*entropy_in07Computing information gain for inv_nodes
gain.inv_nodes <- info.before - info.inv_nodes
gain.inv_nodes## [1] 0.2105958Calculating gain based on node_caps
Looking at distribution in terms of three subsets by node_caps
subsets.node_caps <- table(breast_cancer$node_caps, breast_cancer$class)
subsets.node_caps##
## no-recurrence-events recurrence-events
## no 171 50
## yes 25 31Computing entropy of node_caps subsets
entropy_ncno <-entropy.gini(subsets.node_caps["no",], type=2)$entropy
entropy_ncyes <-entropy.gini(subsets.node_caps["yes",], type=2)$entropy
Computing weighted entropy for node_caps
w_ncno <- sum(subsets.node_caps["no",])/total
w_ncyes <- sum(subsets.node_caps["yes",])/total
info.node_caps <- w_ncno*entropy_ncno + w_ncyes*entropy_ncyesComputing information gain for node_caps
gain.node_caps <- info.before - info.node_caps
gain.node_caps## [1] 0.1840568Calculating gain based on deg_malig
Looking at distribution in terms of three subsets by deg_malig
subsets.deg_malig <- table(breast_cancer$deg_malig, breast_cancer$class)
subsets.deg_malig##
## no-recurrence-events recurrence-events
## 1 57 9
## 2 101 28
## 3 38 44Computing entropy of deg_malig subsets
entropy_dm1 <-entropy.gini(subsets.deg_malig["1",], type=2)$entropy
entropy_dm2 <-entropy.gini(subsets.deg_malig["2",], type=2)$entropy
entropy_dm3 <-entropy.gini(subsets.deg_malig["3",], type=2)$entropy
Computing weighted entropy for deg_malig
w_dm1 <- sum(subsets.deg_malig["1",])/total
w_dm2 <- sum(subsets.deg_malig["2",])/total
w_dm3 <- sum(subsets.deg_malig["3",])/total
info.deg_malig <- w_dm1*entropy_dm1 + w_dm2*entropy_dm2 + w_dm3*entropy_dm3Computing information gain for deg_malig
gain.deg_malig <- info.before - info.deg_malig
gain.deg_malig## [1] 0.2167076Calculating gain based on breast
Looking at distribution in terms of three subsets by breast
subsets.breast <- table(breast_cancer$breast, breast_cancer$class)
subsets.breast##
## no-recurrence-events recurrence-events
## left 100 45
## right 96 36Computing entropy of breast subsets
entropy_left <-entropy.gini(subsets.breast["left",], type=2)$entropy
entropy_right <-entropy.gini(subsets.breast["right",], type=2)$entropy
Computing weighted entropy for breast
w_left <- sum(subsets.breast["left",])/total
w_right <- sum(subsets.breast["right",])/total
info.breast <- w_left*entropy_left + w_right*entropy_rightComputing information gain for breast
gain.breast <- info.before - info.breast
gain.breast## [1] 0.1294075Calculating gain based on breast quadrant
Looking at distribution in terms of three subsets by breast_quad
subsets.breast_quad <- table(breast_cancer$breast_quad, breast_cancer$class)
subsets.breast_quad##
## no-recurrence-events recurrence-events
## central 17 4
## left_low 73 33
## left_up 69 25
## right_low 17 6
## right_up 20 13Computing entropy of breast quadrant subsets
entropy_bq01 <-entropy.gini(subsets.breast_quad["central",], type=2)$entropy
entropy_bq02 <-entropy.gini(subsets.breast_quad["left_low",], type=2)$entropy
entropy_bq03 <-entropy.gini(subsets.breast_quad["left_up",], type=2)$entropy
entropy_bq04 <-entropy.gini(subsets.breast_quad["right_low",], type=2)$entropy
entropy_bq05 <-entropy.gini(subsets.breast_quad["right_up",], type=2)$entropy
Computing weighted entropy for breast quadrant
w_bq01 <- sum(subsets.breast_quad["central",])/total
w_bq02 <- sum(subsets.breast_quad["left_low",])/total
w_bq03 <- sum(subsets.breast_quad["left_up",])/total
w_bq04 <- sum(subsets.breast_quad["right_low",])/total
w_bq05 <- sum(subsets.breast_quad["right_up",])/total
info.breast_quad <- w_bq01*entropy_bq01 + w_bq02*entropy_bq02 + w_bq03*entropy_bq03 + w_bq04*entropy_bq04 + w_bq05*entropy_bq05Computing information gain for breast quadrant
gain.breast_quad <- info.before - info.breast_quad
gain.breast_quad## [1] 0.1368161Calculating gain based on irradiation
Looking at distribution in terms of three subsets by irradiation
subsets.irradiat <- table(breast_cancer$irradiat, breast_cancer$class)
subsets.irradiat##
## no-recurrence-events recurrence-events
## no 164 51
## yes 32 30Computing entropy of irradiation subsets
entropy_irrno <-entropy.gini(subsets.irradiat["no",], type=2)$entropy
entropy_irryes <-entropy.gini(subsets.irradiat["yes",], type=2)$entropy
Computing weighted entropy for irradiation
w_irrno <- sum(subsets.irradiat["no",])/total
w_irryes <- sum(subsets.irradiat["yes",])/total
info.irradiat <- w_irrno*entropy_irrno + w_irryes*entropy_irryesComputing information gain for irradiation
gain.irradiat <- info.before - info.irradiat
gain.irradiat## [1] 0.1628784Results: Choosing root node
# Creating vector of gains for evaluation
gains <- c(gain.age, gain.breast, gain.breast_quad, gain.deg_malig, gain.inv_nodes, gain.irradiat, gain.menopause, gain.node_caps, gain.tumor_size)
# Creating vector of names
gain_names <- c("age", "breast", "breast_quad", "deg_malig", "inv_nodes", "irradiat", "menopause", "node_caps", "tumor_size")
# Creating named vector of gains
names(gains) <- gain_names
# Calculating maximum information gain
# The attribute with the maximum information gain, which will serve as the root node in our decision tree, is:
names(gains)[which.max(gains)]## [1] "deg_malig"1.6.2 Fitting a single tree model using rpart
Decision tree using default settings
set.seed(20)
fit <- rpart(deg_malig~age+breast+breast_quad+inv_nodes+irradiat+menopause+node_caps+tumor_size,
method="class", data=breast_cancer,
control=rpart.control(),
parms=list(split='information'))
summary(fit)## Call:
## rpart(formula = deg_malig ~ age + breast + breast_quad + inv_nodes +
## irradiat + menopause + node_caps + tumor_size, data = breast_cancer,
## method = "class", parms = list(split = "information"), control = rpart.control())
## n= 277
##
## CP nsplit rel error xerror xstd
## 1 0.04391892 0 1.0000000 1.000000 0.05609499
## 2 0.02027027 2 0.9121622 1.020270 0.05599795
## 3 0.01351351 3 0.8918919 1.114865 0.05518860
## 4 0.01126126 10 0.7905405 1.168919 0.05445505
## 5 0.01000000 13 0.7567568 1.148649 0.05475386
##
## Variable importance
## inv_nodes tumor_size node_caps age menopause breast_quad
## 26 23 15 12 9 9
## breast irradiat
## 5 2
##
## Node number 1: 277 observations, complexity param=0.04391892
## predicted class=2 expected loss=0.534296 P(node) =1
## class counts: 66 129 82
## probabilities: 0.238 0.466 0.296
## left son=2 (209 obs) right son=3 (68 obs)
## Primary splits:
## inv_nodes splits as LRRRRRR, improve=20.974280, (0 missing)
## node_caps splits as LR, improve=20.760610, (0 missing)
## tumor_size splits as LLRRRRRRRLR, improve= 9.751465, (0 missing)
## irradiat splits as LR, improve= 9.098196, (0 missing)
## menopause splits as RRL, improve= 2.881789, (0 missing)
## Surrogate splits:
## node_caps splits as LR, agree=0.892, adj=0.559, (0 split)
## irradiat splits as LR, agree=0.769, adj=0.059, (0 split)
##
## Node number 2: 209 observations, complexity param=0.01351351
## predicted class=2 expected loss=0.5358852 P(node) =0.7545126
## class counts: 65 97 47
## probabilities: 0.311 0.464 0.225
## left son=4 (73 obs) right son=5 (136 obs)
## Primary splits:
## tumor_size splits as LLLRRRRRLLL, improve=7.481480, (0 missing)
## node_caps splits as LR, improve=3.733983, (0 missing)
## breast_quad splits as LLRLR, improve=3.080368, (0 missing)
## irradiat splits as LR, improve=3.035000, (0 missing)
## menopause splits as RRL, improve=1.196347, (0 missing)
## Surrogate splits:
## breast_quad splits as LRRRR, agree=0.660, adj=0.027, (0 split)
## menopause splits as RLR, agree=0.656, adj=0.014, (0 split)
##
## Node number 3: 68 observations, complexity param=0.04391892
## predicted class=3 expected loss=0.4852941 P(node) =0.2454874
## class counts: 1 32 35
## probabilities: 0.015 0.471 0.515
## left son=6 (36 obs) right son=7 (32 obs)
## Primary splits:
## tumor_size splits as -RRLRLRRR-L, improve=4.914226, (0 missing)
## inv_nodes splits as -RRRLLL, improve=3.993382, (0 missing)
## menopause splits as R-L, improve=2.885781, (0 missing)
## breast splits as RL, improve=2.874078, (0 missing)
## breast_quad splits as LRLRL, improve=2.874078, (0 missing)
## Surrogate splits:
## inv_nodes splits as -RRLLRR, agree=0.632, adj=0.219, (0 split)
## irradiat splits as LR, agree=0.603, adj=0.156, (0 split)
## age splits as -LLRR-, agree=0.559, adj=0.063, (0 split)
## node_caps splits as LR, agree=0.544, adj=0.031, (0 split)
##
## Node number 4: 73 observations, complexity param=0.01126126
## predicted class=2 expected loss=0.4657534 P(node) =0.2635379
## class counts: 28 39 6
## probabilities: 0.384 0.534 0.082
## left son=8 (42 obs) right son=9 (31 obs)
## Primary splits:
## breast_quad splits as RLRLL, improve=2.319334, (0 missing)
## tumor_size splits as RLR-----LLR, improve=2.233882, (0 missing)
## irradiat splits as LR, improve=1.246302, (0 missing)
## age splits as -LRLLL, improve=1.116546, (0 missing)
## menopause splits as LRR, improve=1.096317, (0 missing)
## Surrogate splits:
## breast splits as LR, agree=0.630, adj=0.129, (0 split)
## node_caps splits as LR, agree=0.616, adj=0.097, (0 split)
## tumor_size splits as RLL-----LLL, agree=0.603, adj=0.065, (0 split)
## age splits as -LLLRL, agree=0.589, adj=0.032, (0 split)
## irradiat splits as LR, agree=0.589, adj=0.032, (0 split)
##
## Node number 5: 136 observations, complexity param=0.01351351
## predicted class=2 expected loss=0.5735294 P(node) =0.4909747
## class counts: 37 58 41
## probabilities: 0.272 0.426 0.301
## left son=10 (87 obs) right son=11 (49 obs)
## Primary splits:
## tumor_size splits as ---LLRRL---, improve=2.4450710, (0 missing)
## irradiat splits as LR, improve=2.4340000, (0 missing)
## menopause splits as RRL, improve=1.5358650, (0 missing)
## breast_quad splits as RRRLR, improve=1.3153470, (0 missing)
## age splits as LRLRRR, improve=0.9967619, (0 missing)
## Surrogate splits:
## menopause splits as LRL, agree=0.654, adj=0.041, (0 split)
## node_caps splits as LR, agree=0.654, adj=0.041, (0 split)
## age splits as RLLLLL, agree=0.647, adj=0.020, (0 split)
## breast_quad splits as LLLLR, agree=0.647, adj=0.020, (0 split)
##
## Node number 6: 36 observations, complexity param=0.02027027
## predicted class=2 expected loss=0.3611111 P(node) =0.1299639
## class counts: 0 23 13
## probabilities: 0.000 0.639 0.361
## left son=12 (21 obs) right son=13 (15 obs)
## Primary splits:
## menopause splits as R-L, improve=3.2256310, (0 missing)
## tumor_size splits as ---L-R----L, improve=1.5076650, (0 missing)
## breast_quad splits as LRRRL, improve=1.0070870, (0 missing)
## breast splits as RL, improve=0.5455044, (0 missing)
## age splits as -LLRR-, improve=0.3636598, (0 missing)
## Surrogate splits:
## age splits as -LLRR-, agree=0.861, adj=0.667, (0 split)
## breast_quad splits as LLLRL, agree=0.667, adj=0.200, (0 split)
## breast splits as RL, agree=0.639, adj=0.133, (0 split)
## inv_nodes splits as -LLRLLL, agree=0.611, adj=0.067, (0 split)
##
## Node number 7: 32 observations
## predicted class=3 expected loss=0.3125 P(node) =0.1155235
## class counts: 1 9 22
## probabilities: 0.031 0.281 0.688
##
## Node number 8: 42 observations, complexity param=0.01126126
## predicted class=2 expected loss=0.4285714 P(node) =0.1516245
## class counts: 17 24 1
## probabilities: 0.405 0.571 0.024
## left son=16 (17 obs) right son=17 (25 obs)
## Primary splits:
## age splits as -LRRLL, improve=2.3409030, (0 missing)
## tumor_size splits as LLR-----LRR, improve=1.6696220, (0 missing)
## menopause splits as LLR, improve=0.7583689, (0 missing)
## breast_quad splits as -R-LL, improve=0.5940572, (0 missing)
## breast splits as LR, improve=0.5723340, (0 missing)
## Surrogate splits:
## menopause splits as LRR, agree=0.643, adj=0.118, (0 split)
## breast_quad splits as -R-LR, agree=0.619, adj=0.059, (0 split)
##
## Node number 9: 31 observations, complexity param=0.01126126
## predicted class=2 expected loss=0.516129 P(node) =0.1119134
## class counts: 11 15 5
## probabilities: 0.355 0.484 0.161
## left son=18 (12 obs) right son=19 (19 obs)
## Primary splits:
## tumor_size splits as RLR------LR, improve=3.9323710, (0 missing)
## age splits as -RRLRR, improve=2.3978190, (0 missing)
## menopause splits as LRR, improve=1.7459430, (0 missing)
## breast splits as LR, improve=0.3546232, (0 missing)
## breast_quad splits as L-R--, improve=0.2327571, (0 missing)
## Surrogate splits:
## age splits as -RLRRL, agree=0.677, adj=0.167, (0 split)
## breast splits as LR, agree=0.645, adj=0.083, (0 split)
##
## Node number 10: 87 observations, complexity param=0.01351351
## predicted class=2 expected loss=0.5057471 P(node) =0.3140794
## class counts: 22 43 22
## probabilities: 0.253 0.494 0.253
## left son=20 (50 obs) right son=21 (37 obs)
## Primary splits:
## menopause splits as R-L, improve=4.0142610, (0 missing)
## breast_quad splits as LLRLR, improve=2.5566890, (0 missing)
## irradiat splits as LR, improve=2.1960260, (0 missing)
## age splits as -LLLRR, improve=1.3840990, (0 missing)
## breast splits as RL, improve=0.4670241, (0 missing)
## Surrogate splits:
## age splits as -LLRRR, agree=0.816, adj=0.568, (0 split)
## breast_quad splits as LLRLL, agree=0.609, adj=0.081, (0 split)
## tumor_size splits as ---RL--L---, agree=0.598, adj=0.054, (0 split)
##
## Node number 11: 49 observations, complexity param=0.01351351
## predicted class=3 expected loss=0.6122449 P(node) =0.1768953
## class counts: 15 15 19
## probabilities: 0.306 0.306 0.388
## left son=22 (30 obs) right son=23 (19 obs)
## Primary splits:
## breast_quad splits as RRLLL, improve=1.7055780, (0 missing)
## irradiat splits as LR, improve=1.2179070, (0 missing)
## tumor_size splits as -----LR----, improve=1.0609000, (0 missing)
## age splits as LRRRL-, improve=0.7532395, (0 missing)
## menopause splits as LRR, improve=0.4104658, (0 missing)
## Surrogate splits:
## breast splits as RL, agree=0.653, adj=0.105, (0 split)
##
## Node number 12: 21 observations
## predicted class=2 expected loss=0.1904762 P(node) =0.07581227
## class counts: 0 17 4
## probabilities: 0.000 0.810 0.190
##
## Node number 13: 15 observations
## predicted class=3 expected loss=0.4 P(node) =0.05415162
## class counts: 0 6 9
## probabilities: 0.000 0.400 0.600
##
## Node number 16: 17 observations
## predicted class=1 expected loss=0.4117647 P(node) =0.06137184
## class counts: 10 7 0
## probabilities: 0.588 0.412 0.000
##
## Node number 17: 25 observations
## predicted class=2 expected loss=0.32 P(node) =0.09025271
## class counts: 7 17 1
## probabilities: 0.280 0.680 0.040
##
## Node number 18: 12 observations
## predicted class=1 expected loss=0.4166667 P(node) =0.0433213
## class counts: 7 5 0
## probabilities: 0.583 0.417 0.000
##
## Node number 19: 19 observations
## predicted class=2 expected loss=0.4736842 P(node) =0.06859206
## class counts: 4 10 5
## probabilities: 0.211 0.526 0.263
##
## Node number 20: 50 observations
## predicted class=2 expected loss=0.42 P(node) =0.1805054
## class counts: 14 29 7
## probabilities: 0.280 0.580 0.140
##
## Node number 21: 37 observations, complexity param=0.01351351
## predicted class=3 expected loss=0.5945946 P(node) =0.133574
## class counts: 8 14 15
## probabilities: 0.216 0.378 0.405
## left son=42 (14 obs) right son=43 (23 obs)
## Primary splits:
## breast_quad splits as LLRRR, improve=1.7855190, (0 missing)
## breast splits as RL, improve=0.8710870, (0 missing)
## age splits as --RRLL, improve=0.7952993, (0 missing)
## tumor_size splits as ---RR--L---, improve=0.2003243, (0 missing)
## Surrogate splits:
## age splits as --RRLR, agree=0.649, adj=0.071, (0 split)
##
## Node number 22: 30 observations, complexity param=0.01351351
## predicted class=2 expected loss=0.6 P(node) =0.1083032
## class counts: 8 12 10
## probabilities: 0.267 0.400 0.333
## left son=44 (9 obs) right son=45 (21 obs)
## Primary splits:
## age splits as LLRRL-, improve=1.6322740, (0 missing)
## menopause splits as LRR, improve=0.9080533, (0 missing)
## breast_quad splits as --LRR, improve=0.6437860, (0 missing)
## tumor_size splits as -----LR----, improve=0.6418279, (0 missing)
## breast splits as LR, improve=0.4714349, (0 missing)
##
## Node number 23: 19 observations
## predicted class=3 expected loss=0.5263158 P(node) =0.06859206
## class counts: 7 3 9
## probabilities: 0.368 0.158 0.474
##
## Node number 42: 14 observations
## predicted class=2 expected loss=0.5 P(node) =0.05054152
## class counts: 4 7 3
## probabilities: 0.286 0.500 0.214
##
## Node number 43: 23 observations, complexity param=0.01351351
## predicted class=3 expected loss=0.4782609 P(node) =0.08303249
## class counts: 4 7 12
## probabilities: 0.174 0.304 0.522
## left son=86 (11 obs) right son=87 (12 obs)
## Primary splits:
## breast splits as RL, improve=2.735390, (0 missing)
## tumor_size splits as ---LR--R---, improve=2.650165, (0 missing)
## age splits as --RRLL, improve=1.319145, (0 missing)
## Surrogate splits:
## breast_quad splits as --LRR, agree=0.609, adj=0.182, (0 split)
## age splits as --RRRL, agree=0.565, adj=0.091, (0 split)
## node_caps splits as RL, agree=0.565, adj=0.091, (0 split)
## tumor_size splits as ---RR--L---, agree=0.565, adj=0.091, (0 split)
##
## Node number 44: 9 observations
## predicted class=2 expected loss=0.4444444 P(node) =0.03249097
## class counts: 3 5 1
## probabilities: 0.333 0.556 0.111
##
## Node number 45: 21 observations
## predicted class=3 expected loss=0.5714286 P(node) =0.07581227
## class counts: 5 7 9
## probabilities: 0.238 0.333 0.429
##
## Node number 86: 11 observations
## predicted class=2 expected loss=0.5454545 P(node) =0.03971119
## class counts: 3 5 3
## probabilities: 0.273 0.455 0.273
##
## Node number 87: 12 observations
## predicted class=3 expected loss=0.25 P(node) =0.0433213
## class counts: 1 2 9
## probabilities: 0.083 0.167 0.750rpart.plot(fit,type=4,extra=2,roundint=FALSE)
Decision tree using custom settings
set.seed(20)
fit <- rpart(deg_malig~age+breast+breast_quad+inv_nodes+irradiat+menopause+node_caps+tumor_size,
method="class", data=breast_cancer,
control=rpart.control(minsplit=4,cp=0.01),
parms=list(split='information'))
summary(fit)## Call:
## rpart(formula = deg_malig ~ age + breast + breast_quad + inv_nodes +
## irradiat + menopause + node_caps + tumor_size, data = breast_cancer,
## method = "class", parms = list(split = "information"), control = rpart.control(minsplit = 4,
## cp = 0.01))
## n= 277
##
## CP nsplit rel error xerror xstd
## 1 0.02702703 0 1.0000000 1.000000 0.05609499
## 2 0.01351351 4 0.8918919 1.013514 0.05603326
## 3 0.01216216 9 0.8243243 1.074324 0.05560809
## 4 0.01126126 14 0.7635135 1.121622 0.05510791
## 5 0.01013514 18 0.7162162 1.121622 0.05510791
## 6 0.01000000 20 0.6959459 1.121622 0.05510791
##
## Variable importance
## tumor_size inv_nodes breast_quad node_caps age breast
## 27 21 13 11 10 8
## irradiat menopause
## 5 5
##
## Node number 1: 277 observations, complexity param=0.02702703
## predicted class=2 expected loss=0.534296 P(node) =1
## class counts: 66 129 82
## probabilities: 0.238 0.466 0.296
## left son=2 (209 obs) right son=3 (68 obs)
## Primary splits:
## inv_nodes splits as LRRRRRR, improve=20.974280, (0 missing)
## node_caps splits as LR, improve=20.760610, (0 missing)
## tumor_size splits as LLRRRRRRRLR, improve= 9.751465, (0 missing)
## irradiat splits as LR, improve= 9.098196, (0 missing)
## menopause splits as RRL, improve= 2.881789, (0 missing)
## Surrogate splits:
## node_caps splits as LR, agree=0.892, adj=0.559, (0 split)
## irradiat splits as LR, agree=0.769, adj=0.059, (0 split)
##
## Node number 2: 209 observations, complexity param=0.01351351
## predicted class=2 expected loss=0.5358852 P(node) =0.7545126
## class counts: 65 97 47
## probabilities: 0.311 0.464 0.225
## left son=4 (73 obs) right son=5 (136 obs)
## Primary splits:
## tumor_size splits as LLLRRRRRLLL, improve=7.481480, (0 missing)
## node_caps splits as LR, improve=3.733983, (0 missing)
## breast_quad splits as LLRLR, improve=3.080368, (0 missing)
## irradiat splits as LR, improve=3.035000, (0 missing)
## menopause splits as RRL, improve=1.196347, (0 missing)
## Surrogate splits:
## breast_quad splits as LRRRR, agree=0.660, adj=0.027, (0 split)
## menopause splits as RLR, agree=0.656, adj=0.014, (0 split)
##
## Node number 3: 68 observations, complexity param=0.02702703
## predicted class=3 expected loss=0.4852941 P(node) =0.2454874
## class counts: 1 32 35
## probabilities: 0.015 0.471 0.515
## left son=6 (62 obs) right son=7 (6 obs)
## Primary splits:
## tumor_size splits as -RRLLLLLR-L, improve=5.939564, (0 missing)
## inv_nodes splits as -RRRLLL, improve=3.993382, (0 missing)
## menopause splits as R-L, improve=2.885781, (0 missing)
## breast splits as RL, improve=2.874078, (0 missing)
## breast_quad splits as LRLRL, improve=2.874078, (0 missing)
##
## Node number 4: 73 observations, complexity param=0.01126126
## predicted class=2 expected loss=0.4657534 P(node) =0.2635379
## class counts: 28 39 6
## probabilities: 0.384 0.534 0.082
## left son=8 (42 obs) right son=9 (31 obs)
## Primary splits:
## breast_quad splits as RLRLL, improve=2.319334, (0 missing)
## tumor_size splits as RLR-----LLR, improve=2.233882, (0 missing)
## node_caps splits as LR, improve=1.921021, (0 missing)
## irradiat splits as LR, improve=1.246302, (0 missing)
## age splits as -LRLLL, improve=1.116546, (0 missing)
## Surrogate splits:
## breast splits as LR, agree=0.630, adj=0.129, (0 split)
## node_caps splits as LR, agree=0.616, adj=0.097, (0 split)
## tumor_size splits as RLL-----LLL, agree=0.603, adj=0.065, (0 split)
## age splits as -LLLRL, agree=0.589, adj=0.032, (0 split)
## irradiat splits as LR, agree=0.589, adj=0.032, (0 split)
##
## Node number 5: 136 observations, complexity param=0.01351351
## predicted class=2 expected loss=0.5735294 P(node) =0.4909747
## class counts: 37 58 41
## probabilities: 0.272 0.426 0.301
## left son=10 (87 obs) right son=11 (49 obs)
## Primary splits:
## tumor_size splits as ---LLRRL---, improve=2.445071, (0 missing)
## irradiat splits as LR, improve=2.434000, (0 missing)
## node_caps splits as LR, improve=2.052153, (0 missing)
## age splits as RRRRRL, improve=1.927031, (0 missing)
## menopause splits as RRL, improve=1.535865, (0 missing)
## Surrogate splits:
## menopause splits as LRL, agree=0.654, adj=0.041, (0 split)
## node_caps splits as LR, agree=0.654, adj=0.041, (0 split)
## age splits as RLLLLL, agree=0.647, adj=0.020, (0 split)
## breast_quad splits as LLLLR, agree=0.647, adj=0.020, (0 split)
##
## Node number 6: 62 observations, complexity param=0.02702703
## predicted class=2 expected loss=0.483871 P(node) =0.2238267
## class counts: 0 32 30
## probabilities: 0.000 0.516 0.484
## left son=12 (15 obs) right son=13 (47 obs)
## Primary splits:
## tumor_size splits as ---LRRRR--L, improve=3.382129, (0 missing)
## inv_nodes splits as -RRRLLL, improve=3.093532, (0 missing)
## menopause splits as R-L, improve=2.619234, (0 missing)
## breast_quad splits as LRLRL, improve=2.090369, (0 missing)
## breast splits as RL, improve=1.582488, (0 missing)
## Surrogate splits:
## inv_nodes splits as -RRLRRR, agree=0.774, adj=0.067, (0 split)
##
## Node number 7: 6 observations
## predicted class=3 expected loss=0.1666667 P(node) =0.02166065
## class counts: 1 0 5
## probabilities: 0.167 0.000 0.833
##
## Node number 8: 42 observations, complexity param=0.01126126
## predicted class=2 expected loss=0.4285714 P(node) =0.1516245
## class counts: 17 24 1
## probabilities: 0.405 0.571 0.024
## left son=16 (17 obs) right son=17 (25 obs)
## Primary splits:
## age splits as -LRRLL, improve=2.3409030, (0 missing)
## tumor_size splits as LLR-----LRR, improve=1.6696220, (0 missing)
## menopause splits as LLR, improve=0.7583689, (0 missing)
## breast_quad splits as -R-LL, improve=0.5940572, (0 missing)
## breast splits as LR, improve=0.5723340, (0 missing)
## Surrogate splits:
## menopause splits as LRR, agree=0.643, adj=0.118, (0 split)
## breast_quad splits as -R-LR, agree=0.619, adj=0.059, (0 split)
##
## Node number 9: 31 observations, complexity param=0.01126126
## predicted class=2 expected loss=0.516129 P(node) =0.1119134
## class counts: 11 15 5
## probabilities: 0.355 0.484 0.161
## left son=18 (12 obs) right son=19 (19 obs)
## Primary splits:
## tumor_size splits as RLR------LR, improve=3.932371, (0 missing)
## irradiat splits as LR, improve=3.181426, (0 missing)
## age splits as -RRLRR, improve=2.397819, (0 missing)
## menopause splits as LRL, improve=1.915670, (0 missing)
## node_caps splits as LR, improve=1.463911, (0 missing)
## Surrogate splits:
## age splits as -RLRRL, agree=0.677, adj=0.167, (0 split)
## breast splits as LR, agree=0.645, adj=0.083, (0 split)
##
## Node number 10: 87 observations, complexity param=0.01216216
## predicted class=2 expected loss=0.5057471 P(node) =0.3140794
## class counts: 22 43 22
## probabilities: 0.253 0.494 0.253
## left son=20 (50 obs) right son=21 (37 obs)
## Primary splits:
## menopause splits as R-L, improve=4.014261, (0 missing)
## breast_quad splits as LLRLR, improve=2.556689, (0 missing)
## age splits as -RRRRL, improve=2.279601, (0 missing)
## irradiat splits as LR, improve=2.196026, (0 missing)
## node_caps splits as RL, improve=1.433499, (0 missing)
## Surrogate splits:
## age splits as -LLRRR, agree=0.816, adj=0.568, (0 split)
## breast_quad splits as LLRLL, agree=0.609, adj=0.081, (0 split)
## tumor_size splits as ---RL--L---, agree=0.598, adj=0.054, (0 split)
##
## Node number 11: 49 observations, complexity param=0.01351351
## predicted class=3 expected loss=0.6122449 P(node) =0.1768953
## class counts: 15 15 19
## probabilities: 0.306 0.306 0.388
## left son=22 (2 obs) right son=23 (47 obs)
## Primary splits:
## breast_quad splits as RRRLR, improve=2.465846, (0 missing)
## node_caps splits as LR, improve=1.893171, (0 missing)
## irradiat splits as LR, improve=1.217907, (0 missing)
## age splits as LRRRR-, improve=1.207596, (0 missing)
## tumor_size splits as -----LR----, improve=1.060900, (0 missing)
##
## Node number 12: 15 observations
## predicted class=2 expected loss=0.2 P(node) =0.05415162
## class counts: 0 12 3
## probabilities: 0.000 0.800 0.200
##
## Node number 13: 47 observations, complexity param=0.02702703
## predicted class=3 expected loss=0.4255319 P(node) =0.1696751
## class counts: 0 20 27
## probabilities: 0.000 0.426 0.574
## left son=26 (24 obs) right son=27 (23 obs)
## Primary splits:
## breast_quad splits as LRLRL, improve=2.5528750, (0 missing)
## inv_nodes splits as -RR-LLL, improve=1.5080160, (0 missing)
## breast splits as RL, improve=1.2247570, (0 missing)
## age splits as -RLRL-, improve=0.7515622, (0 missing)
## tumor_size splits as ----RLRR---, improve=0.7515622, (0 missing)
## Surrogate splits:
## breast splits as RL, agree=0.872, adj=0.739, (0 split)
## age splits as -LLRL-, agree=0.681, adj=0.348, (0 split)
## menopause splits as R-L, agree=0.660, adj=0.304, (0 split)
## node_caps splits as LR, agree=0.596, adj=0.174, (0 split)
## inv_nodes splits as -LL-LRL, agree=0.574, adj=0.130, (0 split)
##
## Node number 16: 17 observations, complexity param=0.01126126
## predicted class=1 expected loss=0.4117647 P(node) =0.06137184
## class counts: 10 7 0
## probabilities: 0.588 0.412 0.000
## left son=32 (15 obs) right son=33 (2 obs)
## Primary splits:
## tumor_size splits as LLL-----LRR, improve=1.9696920, (0 missing)
## breast_quad splits as -R-RL, improve=1.1535550, (0 missing)
## breast splits as RL, improve=0.6976266, (0 missing)
## age splits as -R--RL, improve=0.5523776, (0 missing)
## irradiat splits as RL, improve=0.5523776, (0 missing)
##
## Node number 17: 25 observations
## predicted class=2 expected loss=0.32 P(node) =0.09025271
## class counts: 7 17 1
## probabilities: 0.280 0.680 0.040
##
## Node number 18: 12 observations
## predicted class=1 expected loss=0.4166667 P(node) =0.0433213
## class counts: 7 5 0
## probabilities: 0.583 0.417 0.000
##
## Node number 19: 19 observations
## predicted class=2 expected loss=0.4736842 P(node) =0.06859206
## class counts: 4 10 5
## probabilities: 0.211 0.526 0.263
##
## Node number 20: 50 observations, complexity param=0.01013514
## predicted class=2 expected loss=0.42 P(node) =0.1805054
## class counts: 14 29 7
## probabilities: 0.280 0.580 0.140
## left son=40 (42 obs) right son=41 (8 obs)
## Primary splits:
## irradiat splits as LR, improve=3.9554440, (0 missing)
## breast_quad splits as RRRLR, improve=1.4269000, (0 missing)
## tumor_size splits as ---RR--L---, improve=1.3395510, (0 missing)
## node_caps splits as LR, improve=0.5521029, (0 missing)
## age splits as -RLL--, improve=0.3090612, (0 missing)
##
## Node number 21: 37 observations, complexity param=0.01216216
## predicted class=3 expected loss=0.5945946 P(node) =0.133574
## class counts: 8 14 15
## probabilities: 0.216 0.378 0.405
## left son=42 (32 obs) right son=43 (5 obs)
## Primary splits:
## breast_quad splits as LLLRR, improve=2.6485070, (0 missing)
## age splits as --RRRL, improve=2.2392370, (0 missing)
## node_caps splits as RL, improve=0.9948200, (0 missing)
## irradiat splits as LR, improve=0.9233306, (0 missing)
## breast splits as RL, improve=0.8710870, (0 missing)
## Surrogate splits:
## irradiat splits as LR, agree=0.892, adj=0.2, (0 split)
##
## Node number 22: 2 observations
## predicted class=2 expected loss=0 P(node) =0.007220217
## class counts: 0 2 0
## probabilities: 0.000 1.000 0.000
##
## Node number 23: 47 observations
## predicted class=3 expected loss=0.5957447 P(node) =0.1696751
## class counts: 15 13 19
## probabilities: 0.319 0.277 0.404
##
## Node number 26: 24 observations, complexity param=0.01351351
## predicted class=2 expected loss=0.4166667 P(node) =0.0866426
## class counts: 0 14 10
## probabilities: 0.000 0.583 0.417
## left son=52 (4 obs) right son=53 (20 obs)
## Primary splits:
## age splits as -RRRL-, improve=2.4376950, (0 missing)
## inv_nodes splits as -RR-RLL, improve=1.1694330, (0 missing)
## breast_quad splits as L-R-L, improve=0.7116023, (0 missing)
## irradiat splits as RL, improve=0.3447022, (0 missing)
## breast splits as LR, improve=0.2885215, (0 missing)
## Surrogate splits:
## breast_quad splits as L-R-R, agree=0.875, adj=0.25, (0 split)
##
## Node number 27: 23 observations
## predicted class=3 expected loss=0.2608696 P(node) =0.08303249
## class counts: 0 6 17
## probabilities: 0.000 0.261 0.739
##
## Node number 32: 15 observations
## predicted class=1 expected loss=0.3333333 P(node) =0.05415162
## class counts: 10 5 0
## probabilities: 0.667 0.333 0.000
##
## Node number 33: 2 observations
## predicted class=2 expected loss=0 P(node) =0.007220217
## class counts: 0 2 0
## probabilities: 0.000 1.000 0.000
##
## Node number 40: 42 observations
## predicted class=2 expected loss=0.3809524 P(node) =0.1516245
## class counts: 13 26 3
## probabilities: 0.310 0.619 0.071
##
## Node number 41: 8 observations, complexity param=0.01013514
## predicted class=3 expected loss=0.5 P(node) =0.02888087
## class counts: 1 3 4
## probabilities: 0.125 0.375 0.500
## left son=82 (4 obs) right son=83 (4 obs)
## Primary splits:
## breast splits as LR, improve=3.2958370, (0 missing)
## tumor_size splits as ---RL--L---, improve=1.7260920, (0 missing)
## age splits as -LLR--, improve=0.7648207, (0 missing)
## breast_quad splits as -LR--, improve=0.6103747, (0 missing)
## Surrogate splits:
## age splits as -LRR--, agree=0.750, adj=0.50, (0 split)
## breast_quad splits as -RL--, agree=0.625, adj=0.25, (0 split)
## tumor_size splits as ---RR--L---, agree=0.625, adj=0.25, (0 split)
##
## Node number 42: 32 observations, complexity param=0.01216216
## predicted class=2 expected loss=0.5625 P(node) =0.1155235
## class counts: 6 14 12
## probabilities: 0.188 0.438 0.375
## left son=84 (14 obs) right son=85 (18 obs)
## Primary splits:
## breast_quad splits as LLR--, improve=1.6588870, (0 missing)
## age splits as --RRRL, improve=1.3359340, (0 missing)
## node_caps splits as RL, improve=0.8474846, (0 missing)
## tumor_size splits as ---LR--L---, improve=0.4361799, (0 missing)
## breast splits as RL, improve=0.3107940, (0 missing)
## Surrogate splits:
## tumor_size splits as ---RL--R---, agree=0.656, adj=0.214, (0 split)
## age splits as --RRLR, agree=0.625, adj=0.143, (0 split)
##
## Node number 43: 5 observations
## predicted class=3 expected loss=0.4 P(node) =0.01805054
## class counts: 2 0 3
## probabilities: 0.400 0.000 0.600
##
## Node number 52: 4 observations
## predicted class=2 expected loss=0 P(node) =0.01444043
## class counts: 0 4 0
## probabilities: 0.000 1.000 0.000
##
## Node number 53: 20 observations, complexity param=0.01351351
## predicted class=2 expected loss=0.5 P(node) =0.07220217
## class counts: 0 10 10
## probabilities: 0.000 0.500 0.500
## left son=106 (8 obs) right son=107 (12 obs)
## Primary splits:
## inv_nodes splits as -RR-RLL, improve=1.7260920, (0 missing)
## age splits as -LLR--, improve=1.2657560, (0 missing)
## menopause splits as R-L, improve=0.6485757, (0 missing)
## tumor_size splits as ----RLLR---, improve=0.4201185, (0 missing)
## irradiat splits as RL, improve=0.4027103, (0 missing)
## Surrogate splits:
## age splits as -LRR--, agree=0.65, adj=0.125, (0 split)
## tumor_size splits as ----RRLR---, agree=0.65, adj=0.125, (0 split)
##
## Node number 82: 4 observations
## predicted class=2 expected loss=0.25 P(node) =0.01444043
## class counts: 0 3 1
## probabilities: 0.000 0.750 0.250
##
## Node number 83: 4 observations
## predicted class=3 expected loss=0.25 P(node) =0.01444043
## class counts: 1 0 3
## probabilities: 0.250 0.000 0.750
##
## Node number 84: 14 observations, complexity param=0.01216216
## predicted class=2 expected loss=0.5 P(node) =0.05054152
## class counts: 4 7 3
## probabilities: 0.286 0.500 0.214
## left son=168 (6 obs) right son=169 (8 obs)
## Primary splits:
## tumor_size splits as ---LR--R---, improve=2.5310160, (0 missing)
## age splits as --RLR-, improve=1.5716460, (0 missing)
## breast_quad splits as RL---, improve=0.7316467, (0 missing)
## breast splits as LR, improve=0.6214736, (0 missing)
## Surrogate splits:
## breast_quad splits as LR---, agree=0.643, adj=0.167, (0 split)
##
## Node number 85: 18 observations, complexity param=0.01216216
## predicted class=3 expected loss=0.5 P(node) =0.06498195
## class counts: 2 7 9
## probabilities: 0.111 0.389 0.500
## left son=170 (10 obs) right son=171 (8 obs)
## Primary splits:
## breast splits as RL, improve=2.4487940, (0 missing)
## age splits as --RRLL, improve=2.1119580, (0 missing)
## tumor_size splits as ---RR--L---, improve=1.9564520, (0 missing)
## node_caps splits as RL, improve=0.9912506, (0 missing)
##
## Node number 106: 8 observations
## predicted class=2 expected loss=0.25 P(node) =0.02888087
## class counts: 0 6 2
## probabilities: 0.000 0.750 0.250
##
## Node number 107: 12 observations
## predicted class=3 expected loss=0.3333333 P(node) =0.0433213
## class counts: 0 4 8
## probabilities: 0.000 0.333 0.667
##
## Node number 168: 6 observations
## predicted class=1 expected loss=0.5 P(node) =0.02166065
## class counts: 3 1 2
## probabilities: 0.500 0.167 0.333
##
## Node number 169: 8 observations
## predicted class=2 expected loss=0.25 P(node) =0.02888087
## class counts: 1 6 1
## probabilities: 0.125 0.750 0.125
##
## Node number 170: 10 observations
## predicted class=2 expected loss=0.5 P(node) =0.03610108
## class counts: 2 5 3
## probabilities: 0.200 0.500 0.300
##
## Node number 171: 8 observations
## predicted class=3 expected loss=0.25 P(node) =0.02888087
## class counts: 0 2 6
## probabilities: 0.000 0.250 0.750rpart.plot(fit,type=4,extra=2,roundint=FALSE)
Comparison of decision trees
# The decision tree created with default settings has a height of 7, while the decision tree created with custom settings has a height of 8. The default tree will yield a faster speed of prediction, and will not be as likely to be overfitting of our sample data. However, further testing will likely generate a tree that is less subject to overfitting with a faster speed of prediction.1.6.3 Building a bossted tree with holdout validation
Cleaning data for use in boosted tree generation
# Since the response must be 0 and 1, we mutate the class attribute:
breast_cancer <- breast_cancer %>% mutate(recur_label=ifelse(class=="recurrence-events", 1, 0))Splitting data into two sets for training and testing (holdout validation)
set.seed(52)
# Shuffling row indices
rows <- sample(nrow(breast_cancer))
# Randomly reordering rows of dataset using rows vector.
breast_cancer <- breast_cancer[rows,]
split <- round(nrow(breast_cancer)*0.66)
train <- breast_cancer[1:split, ]
test <- breast_cancer[split+1:nrow(breast_cancer), ]Creating boosted tree
trainFormula <- recur_label~deg_malig+age+breast+breast_quad+inv_nodes+irradiat+menopause+node_caps+tumor_size
trainX <- build.x(trainFormula,train,contrasts=FALSE)
trainY <- build.y(trainFormula,train)
trainBoost <- xgboost::xgboost(
data=trainX,
label=trainY,
max.depth=3,
eta=0.3,
#nthread=4,
nrounds=10,
objective="binary:logistic")## [1] train-error:0.224044
## [2] train-error:0.213115
## [3] train-error:0.218579
## [4] train-error:0.234973
## [5] train-error:0.218579
## [6] train-error:0.196721
## [7] train-error:0.202186
## [8] train-error:0.191257
## [9] train-error:0.191257
## [10] train-error:0.196721xgb.plot.multi.trees(
trainBoost,
feature.names=colnames(trainX)
)## Column 2 ['No'] of item 2 is missing in item 1. Use fill=TRUE to fill with NA (NULL for list columns), or use.names=FALSE to ignore column names. use.names='check' (default from v1.12.2) emits this message and proceeds as if use.names=FALSE for backwards compatibility. See news item 5 in v1.12.2 for options to control this message.Single tree plots
xgb.plot.tree(model=trainBoost, trees=2:3, render=TRUE)1.6.4 Ranking variables in terms of their importance to the boosted tree
Using xgb
importance_matrix <- xgb.importance(
model = trainBoost,
feature_names =colnames(trainX)
)
print(importance_matrix)## Feature Gain Cover Frequency
## 1: deg_malig 0.301240180 0.18227566 0.10909091
## 2: inv_nodes0-2 0.150060532 0.12733374 0.10909091
## 3: tumor_size10-14 0.064638853 0.12540951 0.07272727
## 4: breast_quadleft_low 0.059999214 0.02371011 0.10909091
## 5: node_capsno 0.049320091 0.02367454 0.03636364
## 6: age30-39 0.046434893 0.07822831 0.05454545
## 7: tumor_size30-34 0.044341436 0.03115341 0.07272727
## 8: irradiatno 0.038286389 0.03753707 0.05454545
## 9: breast_quadright_up 0.036663111 0.05114478 0.03636364
## 10: breastleft 0.035637076 0.03344851 0.07272727
## 11: menopausepremeno 0.034758201 0.05609384 0.05454545
## 12: tumor_size20-24 0.030920365 0.05056489 0.05454545
## 13: age60-69 0.029433021 0.01526813 0.03636364
## 14: breast_quadright_low 0.025520765 0.03538208 0.03636364
## 15: tumor_size40-44 0.017492958 0.04838156 0.03636364
## 16: tumor_size35-39 0.016262771 0.02807589 0.01818182
## 17: age50-59 0.011874616 0.02942488 0.01818182
## 18: tumor_size50-54 0.007115528 0.02289308 0.01818182xgb.plot.importance(importance_matrix = importance_matrix)
Using vip
vip::vip(trainBoost,num_features=18)
1.6.5 Constructing random forest model
trainY <- build.y(trainFormula,train)
boostedForest <- xgboost(
data=trainX,
label=trainY,
max_depth=4,
num_parallel_tree=1000,
subsample=0.5,
colsample_bytree=0.5,
nrounds=3,
objective="binary:logistic"
)## [1] train-error:0.224044
## [2] train-error:0.224044
## [3] train-error:0.229508xgb.plot.multi.trees(
boostedForest,
feature.names=colnames(trainX)
)## Column 2 ['No'] of item 2 is missing in item 1. Use fill=TRUE to fill with NA (NULL for list columns), or use.names=FALSE to ignore column names. use.names='check' (default from v1.12.2) emits this message and proceeds as if use.names=FALSE for backwards compatibility. See news item 5 in v1.12.2 for options to control this message.